Non-exponential relaxation for anomalous diffusion

Abstract

We study the relaxation process in normal and anomalous diffusion regimes for systems described by a generalized Langevin equation (GLE). We demonstrate the existence of a very general correlation function which describes the relaxation phenomena. Such function is even; therefore, it cannot be an exponential or a stretched exponential. However, for a proper choice of the parameters, those functions can be reproduced within certain intervals with good precision. We also show the passage from the non-Markovian to the Markovian behaviour in the normal diffusion regime. For times longer than the relaxation time, the correlation function for anomalous diffusion becomes a power law for broadband noise.

Subdiffusive motion of a polymer composed of subdiffusive monomers

“…21 from t to s , rearranging, and inverting, we arrive at the solution $$C_{p}false(tfalse)=\frac{3k_{B}T}{k_p}{E}_{\alpha ,1}true[-\frac{k_p}{N\xi \mathrm{normal\Gamma }false(3-\alpha false)}{t}^{\alpha }true],$$ where E α,β ( x ) is the Mittag-Leffler function $$E_{\alpha ,\beta }false(xfalse)=\sum _{j=0}^{\infty }\frac{x^j}{\mathrm{normal\Gamma }false(\beta +\alpha jfalse)}.$$ In the limit α → 1, the correlation function is C p ( t ) = (3 k B T/k p ) exp [− k p t /( N ξ)], which corresponds to the behavior of the Rouse model in a Newtonian fluid [17]. However, in a viscoelastic fluid, where 0 < α < 1, the correlation function decays more slowly than an exponential, characteristically as a stretched exponential at short times and as an inverse power law at long times [21]. …”

Khinchin Theorem and Anomalous Diffusion

“…where the exponent classifies the different types of diffusion: subdiffusion for 0 < < 1, normal diffusion for ¼ 1, and superdiffusion for 1 < 2; for ¼ 2 the process is called ballistic [5,[14][15][16][17]. According to Kubo's linear response theory [18], the diffusion constant is given by…”

Anomalous diffusion and basic theorems of statistical mechanics

“…Noise of this form can be obtained either by formal methods or empirical data. Using this expression, Vainstein et al [12] have shown that ν = β for β ≤ 1 and ν = 1 for β ≥ 1 consequently ν ≤ 1, i.e. ballistic is the highest asymptotic limit for diffusion.…”

“…where the exponent α classifies the different types of diffusion: subdiffusion for 0 < α < 1, normal diffusion for α = 1, and superdiffusion for 1 < α ≤ 2; for α = 2 the process is called ballistic [5,7,11,12,13]. Morgado et al [11] obtained a general relationship between the Laplace transform of the memory functionΠ(z) and the diffusion exponent…”